Temporal Recurring Unavailabilities in Multi-agent Rural Postman Problem: Navigating railway tracks during availability time intervals

Abstract

Time-dependent (or temporal) properties may arise in many network-based planning problems, particularly in the routing and scheduling of railway track inspection problems. The availability of tracks depends on the train schedules, maintenance possessions, etc. In the absence of side constraints, this routing and scheduling problem is formulated as a multi-agent rural postman problem on a temporal-directed network; where a given set of rail track sections must be visited while respecting the temporal attributes due to railway track unavailabilities. In this work, we adopt a three-index formulation for the multi-agent Rural Postman Problem with Temporal Recurring Unavailabilities (RPP-TRU) and frame it as a Mixed Integer Linear Programming (MILP) problem. In addition, we propose relevant theoretical studies for RPP-TRU to ensure the feasibility of the proposed optimization problem. Two approaches of an exact algorithm are proposed, based on Benders' decomposition framework, to address the disjunctive unavailability constraints occurring in its scheduling sub-problems, alongside the NP-Hard routing (master) problem. A polynomial-time algorithm is designed to address the scheduling sub-problem, while the NP-Hard master problem is solved using MILP toolbox. Comparison results with RPP (without temporal constraints) show a minor compromise with the spatial cost solution with significantly less delay, hence suitable for real-world routing and scheduling applications occurring in a shared network like railways. A simulation study on a part of the Mumbai suburban railway network demonstrates the working of the proposed methodology under a realistic setting.

Figures (16)

• Figure 1: A sub-graph is constructed from a multi-graph by selecting all vertices and arcs strictly inside some region (the shaded blob).
• Figure 2: An example graph from lannez, where vertex $v_1$ is the depot, and $A_R := \{a_2, a_5\}$ represent the service arcs. The running-time data is given as $W = [1,3,1,1,2,1]^\top$ for single-agent case, and $W = [[1,3,1,1,2,1]^\top,[1,3,1,1,2,1]^\top]$ for two-agent case.
• Figure 3: A replicated graph for $|\mathcal{K}|$ agents, constructed from a base graph shown in Figure \ref{['fig:prelim_fig2']}.
• Figure 4: Three arcs of a graph $G$ having different unavailability periods are shown. Based on the running-time data $W$ (illustrated with slanting lines) and the unavailability list $Z$ (fat blocks), the infeasible regions for the departure time of an agent (narrow boxes over dashed lines) at each of the tail vertices of corresponding arcs are computed. This restriction guarantees that for any arc of the replicated graph, if an agent's departure time at the tail vertex is outside the highlighted infeasible region, then the agent will reach the other end without entering the unavailability zone (fat blocks) of the arc and then wait as long as required at the tail vertex of next arc. For instance, if an agent departs at 10:59 am from the tail vertex of arc $a_1$, denoted as $v_{1,k,l}$ then the agent arrives at the head vertex before the arc gets blocked at 12:29 pm. The agent then waits till 3:00 pm as the next arc $a_2$ will be blocked midway of its traversal, if it departs at 12:29 pm.
• Figure 5: Proposed Benders' decomposition algorithm variations for RPP-TRU, showing reduced master problems \ref{['form:rmp1']} and \ref{['form:rmp2']}, and temporal sub-problems \ref{['form:bsp']} for each $k \in \mathcal{K}$. The pseudo-code for TemporalSol and GenUnavCuts is shown in Algorithm \ref{['algo:temporal']} and \ref{['algo:unav']} respectively.

Theorems & Definitions (13)

• Theorem 3.1
• Corollary 3.1.1
• Corollary 3.1.2
• Remark 1
• Remark 2
• Lemma 5.1
• Remark 3
• Corollary 5.1.1
• Remark 4
• Remark 5